what is the reminder when 6^17+17^6 is divided by 7?

• 1) 6^17 mod 7=(7-1)^17 mod 7
  so 7^17 divide by 7 and remainder will 0,so we only take (-1)^17 mod 7=-1
  
  2) 17^6 mod 7=(7*2+3)^6
  so now (7*2)^17 divide by 7 and remainder will 0,so we only take (3)^6 mod 7=
  729 mod 7=1
  now add (1)+(2)
  -1+1=0(ans)
  
• 11 years agoHelpfull: Yes(18) No(1)
• Hi All,
  
  First break the sum into two parts..
  (i) 6^17 (ii) 17^6
  
  sol-(i)- By remainder theorem there are always two remainders for any number, i.e. one is in negative and another is positive. In this case when we divide 6 by 7 we will get two remainders that is 6 and (-1). We can take any of these but for simplicity we will take (-1) is the remainder. Now (-1)^17 = -1.
  
  sol-(ii)-- we have 17^6.. Now. 17^6 /7 = 3^6/7= (3^3)^2/7 = 27^2/7 = (-1)^2/7 = 1 is the remainder.
  
  On adding case -(i) and (ii)... we will get final remainder = 0.
  
  Thanks.
• 11 years agoHelpfull: Yes(14) No(1)
• how did this came : 17^6 /7 = 3^6/7 ,, can you please explain ! @vikas goyal
• 11 years agoHelpfull: Yes(1) No(2)
• the units digit in 6^n is always 6 so the units digit in 6^17 is 6
  the units digit in 17^1=7
  17^2=9
  17^3=3
  17^4=1 and the cycle repeats
  so the units digit in 17^6 =9
  in order to get the remainder of 6^17+17^6 when divided by 7 we have to add the units digits of 6^17 and 17^6 i.e., 9+6=15. the units digit is 5 which is the remainder..
  ans is 5..
• 11 years agoHelpfull: Yes(1) No(3)
• answer is 0
  
• 11 years agoHelpfull: Yes(0) No(1)
• answer is 0
• 11 years agoHelpfull: Yes(0) No(1)
• The answer is 0
• 11 years agoHelpfull: Yes(0) No(2)
• the answer was '0'
• 10 years agoHelpfull: Yes(0) No(1)